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Properties

Label 7T5
Degree $7$
Order $168$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\GL(3,2)$

Related objects

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Group action invariants

Degree $n$:  $7$
Transitive number $t$:  $5$
Group:  $\GL(3,2)$
CHM label:  $L(7) = L(3,2)$
Parity:  $1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (1,2)(3,6), (1,2,3,4,5,6,7)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

7T5, 8T37, 14T10 x 2, 21T14, 24T284, 28T32, 42T37, 42T38 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1 $ $21$ $2$ $(3,5)(6,7)$
$ 4, 2, 1 $ $42$ $4$ $(2,3,4,7)(5,6)$
$ 3, 3, 1 $ $56$ $3$ $(2,3,5)(4,7,6)$
$ 7 $ $24$ $7$ $(1,2,3,4,5,6,7)$
$ 7 $ $24$ $7$ $(1,2,3,7,6,4,5)$

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  [168, 42]
Character table:    
     2  3  3  2  .  .  .
     3  1  .  .  1  .  .
     7  1  .  .  .  1  1

       1a 2a 4a 3a 7a 7b
    2P 1a 1a 2a 3a 7a 7b
    3P 1a 2a 4a 1a 7b 7a
    5P 1a 2a 4a 3a 7b 7a
    7P 1a 2a 4a 3a 1a 1a

X.1     1  1  1  1  1  1
X.2     3 -1  1  .  A /A
X.3     3 -1  1  . /A  A
X.4     6  2  .  . -1 -1
X.5     7 -1 -1  1  .  .
X.6     8  .  . -1  1  1

A = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7

Additional information

This is the lowest degree transitive permutation group which admits a Gassmann triple. As a result, number fields with this Galois group come in arithmetically equivalent pairs.